The need for signal enhancement arises when the original signal carried certain important features. When this signal is passed or transmitted through a band-limited channel such as an imaging system with limited resolving power or band-limited communication channel, or through turbulent atmosphere, the above-mentioned signal features are destroyed or degraded. Typical examples of such features are edges and lines in visual and radar images as well as various speech features.
Prior art solutions typically fall in two distinct classes. The first class is edge crispening achieved through processing the signal of interest with a high frequency bandpass characteristic. This method almost invariably produces overshoots and undershoots of the original singular features such as edges and lines. For some of the earlier uses of this technique, see L. G. Roberts, "Machine Perception of Three-Dimensional Solids," in Optical and Electro-Optical Information Processing, J. T. Tippett et al., eds. MIT Press, Cambridge, Mass., 1965. The major limitations of the high-pass filtering approach follows from its linear nature. First, it produces unacceptable artifacts in the signal such as ringing (see Donald C. Wells, "Nonlinear Image Restoration: What We Have Learned," SPIE, Vol. 264, Applications of Digital Image Processing to Astronomy, 1980, pp. 148-155) and aliasing, which can hopelessly confuse subsequent machine recognition procedures and produce faulty feature detection results. Secondly, and just as important, the high-pass filter will not distinguish between high-frequency contributions caused by the feature, e.g., each discontinuity producing a 1/.omega. asymptotic behavior, from the broad spectrum/noise contribution. Thus, the resulting signal is polluted by the explosion of the noise component. Grunlund, G. M., H. Knutsson and R. Wilson in "Image Enhancement," Fundamentals in Computer Vision, ed. O. J. Fougeros, 1983, pp. 56-57, proposed a filter which is a linear contribution of two components--a linear low-pass filter and a linear high-pass filter. The weights of this combination are determined from the local operation amounting to local edge and line detection. This filter is applied to the problem of edge-preserving noise reduction and represents a first attempt to perform a context-sensitive image denoising. However, since this algorithm is basically a linear combination of the two linear filters, it is bound either to create spurious artifacts in images or to blur significant image features.
The second class of solutions falls in the category of signal restoration, e.g., image deblurring or seismic and audio deconvolution. H. C. Andrews and B. R. Hunt in "Digital Image Restoration," Prentice-Hall, Englewood Cliffs, N.J., 1977, is a definitive source for the most important image restoration algorithms in use in today's image-processing applications. Typically, blurring is modeled by the convolution of the original signal with the impulse response (point spread) function of the degradation system.
Basic deblurring methods assume exact knowledge of the blurring kernel. The deconvolution algorithms include direct methods: inverse filtering, Wiener filtering, power spectral equalization, constrained least-squares; and indirect techniques: maximum a posteriori restoration, maximum entropy restoration. For the maximum entropy technique, see B. R. Frieden, "Restoring With Maximum Likelihood and Maximum Entropy," J. Opt. Soc. Am. 62, 1972, 511-518.
The major drawbacks of the above techniques are similar to the shortcomings of the traditional enhancement techniques. The direct techniques linearly trade off bandwidth of the solution with the stability of the inversion process, since deconvolution is a classically ill-posed problem.
The indirect methods are nonlinear regularization procedures which produce smooth solutions. However, such solutions may be of little interest when the object of deblurring is to restore singular features which are essentially nonsmooth functional behavior such as jumps, corners, etc.
In the class of restoration methods, the closest prior art has been proposed by A. Carasso, J. Sanderson, J. Hyman in "Digital Removal of Random Media Image Degradation by Solving the Diffusion Equation Backwards in Time," SIAM J. Numer. Anal., Vol. 15, No. 2, 1978, pp. 344-367. There, the authors treat the problem of restoration by converting the usual convolution integral equation into the backwards heat equation. Then it is observed that this equation is extremely ill-posed, and certain bounds are imposed on the class of possible solutions. The regularized (satisfying these bounds) solution is approximated by solving a backwards beam equation which is well-posed. The solution reliability estimate shows that only partial restoration is possible. The algorithm slows down the noise explosion by recovering the image backwards in time. Thus the method takes a blurred image and derives a noisier but less blurred image. A return to the original zero time will provide totally redundant information. Also, the regularization bounds do not provide restoration of discontinuities or other singular features (e.g. lines) in an artifact-free manner.
A new stable method or procedure is needed in order to restore singular features of signals without numerical contamination by spurious artifacts and with a controlled order of accuracy of the restored singularities.
The subject of this invention is the development of such a stable feature-oriented procedure and apparatus.